`x^(2)-22x+40` =_______

Options

A. `(x-10)(x+4)`
B. `(x+2)(x-20)`
C. `(x+10)(x-4)`
D. `(x-20)(x-2)`

Answer

To factor the expression \( x^2 - 22x + 40 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -22 \) (coefficient of \( x \))
- \( c = 40 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers \( p \) and \( q \) such that:
- \( p \times q = ac = 1 \times 40 = 40 \)
- \( p + q = b = -22 \)

### Step 3: List the factor pairs of 40
The factor pairs of 40 are:
- \( (1, 40) \)
- \( (2, 20) \)
- \( (4, 10) \)
- \( (5, 8) \)

### Step 4: Check which pair adds to -22
We need to consider negative pairs since we need the sum to be negative:
- \( (-2, -20) \) gives \( -2 + (-20) = -22 \) (this works)
- Other pairs do not yield -22.

### Step 5: Rewrite the middle term using the found factors
We can rewrite the expression \( x^2 - 22x + 40 \) as:
\[ x^2 - 20x - 2x + 40 \]

### Step 6: Factor by grouping
Now, we can group the terms:
\[ (x^2 - 20x) + (-2x + 40) \]

Factoring out the common factors in each group:
\[ x(x - 20) - 2(x - 20) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x - 20) \):
\[ (x - 20)(x - 2) \]

### Final Answer
Thus, the factored form of \( x^2 - 22x + 40 \) is:
\[ (x - 20)(x - 2) \]

---